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Resource allocation games: a compromise stable extension of bankruptcy games. (English) Zbl 1280.91013
This paper presents an extension of the traditional bankruptcy problem $$(N,E,d)$$ to the resource allocation problem $$(N,E,d, \alpha)$$, where $$N$$ represents a finite set of claimants, $$E\geq 0$$ is the estate which has to be divided among the claimants, $$d=(d_i)_{i\in N}$$ is a vector in $$\mathbb R^N_{++}$$ with $$d_i$$ representing agent $$i$$’s claim on the estate ($$\sum_{i\in N}d_i\geq E$$), and $$\alpha=(\alpha_i)_{i\in N}$$ is a vector in $$\mathbb R^N_{++}$$ with $$\alpha_i$$ describing agent $$i$$’s reward function $$r_i(x_i)=\alpha_ix_i$$ (here $$x_i\geq 0$$ is a feasible assignment part possible to be given to agent $$i$$ restricted by: $$0\leq x_i\leq d_i$$ and $$\sum_{i\in N}x_i=E$$). The problem is how to find the “best” feasible assignment vector $$x=(x_i)_{i\in N}$$ among the optimal assignments, that is satisfying $$\sum_{i\in N}\alpha_ix_i =\max$$. To solve this problem, the authors construct a TU-game $$v^R$$ (called a resource allocation game) corresponding to the resource allocation problem $$(N,E,d, \alpha)$$. It is shown for the game $$v^R$$ that it is compromise stable and totally balanced. Furthermore, an explicit expression for the nucleolus of the game $$v^R$$ has been derived.

##### MSC:
 91A12 Cooperative games 91A06 $$n$$-person games, $$n>2$$ 91B32 Resource and cost allocation (including fair division, apportionment, etc.)
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##### References:
 [1] Aumann, R; Maschler, M, Game theoretic analysis of a bankruptcy problem from the talmud, J Econ Theory, 36, 195-213, (1985) · Zbl 0578.90100 [2] Bergantiños G, Lorenzo L, Lorenzo-Freire S (2010) A characterization of the proportional rule in multi-issue allocation situations. Oper Res Lett 38(1):17-19. doi:10.1016/j.orl.2009.10.003. http://www.sciencedirect.com/science/article/pii/S0167637709001187 · Zbl 1198.91094 [3] Borm, P; Hamers, H; Hendrickx, R, Operations research games: a survey, TOP, 9, 139-216, (2001) · Zbl 1006.91009 [4] Calleja, P; Borm, P; Hendrickx, R, Multi-issue allocation situations, Eur J Oper Res, 164, 730-747, (2005) · Zbl 1057.91003 [5] Carraro C, Marchiori C, Sgobbi A (2005) Applications of negotiation theory to water issues. Working Papers 2005.65, Fondazione Eni Enrico Mattei. Extracted from http://econpapers.repec.org/RePEc:fem:femwpa:2005.65 [6] Grundel S, Borm P, Hamers H (2013) Resource allocation games with concave reward functions. Tilburg University, Mimeo · Zbl 1280.91013 [7] Kaminski, M, Hydraulic rationing, Math Soc Sci, 40, 131-155, (2000) · Zbl 0967.90070 [8] Madani K (2010) Game theory and water resources. J Hydrol 381(3):225-238 · Zbl 1086.91008 [9] Moulin H (1991) Axioms of cooperative decision making. Cambridge University Press, Cambridge · Zbl 0699.90001 [10] O’Neill, B, A problem of rights arbitration from the talmud, Math Soc Sci, 2, 345-371, (1982) · Zbl 0489.90090 [11] Parrachino I, Dinar A, Patrone F (2006) Cooperative game theory and its application to natural, environmental, and water resource issues: 3. application to water resources. Policy Research Working Paper Series 4074, The World Bank. Extracted from http://econpapers.repec.org/RePEc:wbk:wbrwps:4074 · Zbl 1057.91003 [12] Potters, J; Tijs, S; Megiddo, N (ed.), On the locus of the nucleolus, 193-203, (1994), Berlin · Zbl 0807.90140 [13] Quant, M; Borm, P; Reijnierse, H; Velzen, S, The core cover in relation to the nucleolus and the Weber set, Int J Game Theory, 33, 491-503, (2005) · Zbl 1086.91008 [14] Ransmeier J (1942) The tennessee valley authority: a case study in the economics of multiple purpose stream planning. Vanderbilt University Press, Nashville [15] Schmeidler, D, The nucleolus of a characteristic function game, SIAM J Appl Math, 17, 1163-1170, (1969) · Zbl 0191.49502 [16] Straffin, P; Heaney, J, Game theory and the tennessee valley authority, Int J Game Theory, 10, 35-43, (1981) · Zbl 0452.90100 [17] Thomson, W, Axiomatic and game theoretic analysis of bankruptcy and taxation problems: a survey, Math Soc Sci, 45, 249-297, (2003) · Zbl 1042.91014 [18] Tijs, S; Lipperts, F, The hypercube and the core cover of $$n$$-person cooperative games, Cahiers du Centre d’Études de Recherche Opérationelle, 24, 27-37, (1982) · Zbl 0479.90093 [19] Young, H, Distributive justice of taxation, J Econ Theory, 44, 321-335, (1988) · Zbl 0637.90027 [20] Young H (1995) Equity, in theory and practice. Princeton University Press, Princeton [21] Zara S, Dinar A, Patrone F (2006) Cooperative game theory and its application to natural, environmental, and water resource issues: 2. application to natural and environmental resources. Policy Research Working Paper Series 4073, The World Bank. Extracted from http://econpapers.repec.org/RePEc:wbk:wbrwps:4073
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